Now, the Mega Millions jackpot has reached a ludicrously high
$636 million, and so we decided to repeat the analysis for this
lottery— ahead
of tonight's drawing is at 11 PM EST.

The most basic
thing to consider in any game of chance, whether you are looking
at a lottery, a roulette wheel, or a deck of cards in a blackjack
game, is yourexpected
winningsfrom the game. This is
basically a measure of how much you will win on average by
playing the game a large number of times.

The expected winnings for a game can be found if we know the
probabilities of the different outcomes in the game, and the
earnings (or losses) associated with each outcome. We multiply
the probabilities by the earnings or losses, and then add these
up.

The expected value will tell us what will happen on average if we
play a game a large number of times. If the expected value is
positive, then on average we are winning money, and so if we play
the game long enough, we should end up in the black. If the
expected value is negative, then on average we are losing money,
and so we should not be playing this game.

We can find the expected value for playing Mega Millions
with the current ridiculous jackpot. In Mega Millions, you pick
five different numbers between 1 and 75, and one number between 1
and 15. If all six match, you win the jackpot. If you match some
of the first five numbers and/or the last number, you win a
smaller prize.

The first column shows each outcome — how many of the five
numbers from 1 to 75 we got right, and whether or not we got the
last number from 1 to 15. The second column shows the prize and
the third the prize less our investment of a dollar per ticket.
In the fourth column, we have the odds of getting each outcome,
as per the Mega Millions website. In the fifth column, we convert
those odds into a probability between zero and one by taking the
reciprocal of the odds. Finally, we multiply together our
winnings less investment by the probability, and sum these up,
giving us the expected value.

Notice that by far our most likely outcome is that we get
none of the numbers right. There is about a 93% chance that we
just wind up losing our dollar. Despite this, since the jackpot
is so enormously high right now, our average winnings are a nice
positive $1.63, indicating that we should consider buying a
ticket.

Another notable property of Mega Millions is that the lower
prizes do not help us too much. Without that jackpot, the
expected value is a very unhappy —$0.82. So, Mega Millions really
is all about the jackpot.

One factor to consider is that, as with most lotteries, there are
two options for the prize: the advertised prize of $636 million
is based on a thirty year annuity, receiving the prize in smaller
annual chunks for thirty years. The other option is to take cash
up front, but at a huge discount. Today's up front cash prize is
$341 million — a big drop, but nothing to sneeze at. We can find
the expected winnings for taking the cash up front:

Business Insider/ Mega
Millions

The expected value of a Mega Millions ticket here is lower — just
$0.49 as opposed to the $1.63 taking the annuity — but it is
still positive, and this is still a viable option.

There are other confounding factors — you also have to consider
the effect of taxes, and how that plays with both options. Even
though this lottery has a positive expected value, it has an
extremely high variance and standard deviation, owing to the fact
that there is one very, very unlikely outcome where you do
incredibly well, and the overwhelming majority of the time, you
just lose a buck.

The most interesting confounding factor is the possibility of
multiple winners. How many people are playing will affect
the odds of a split and the size of the resulting jackpot
(whether it is broken in to two parts, or three, or four, or
however many winners there are). We can, however, estimate the
expected value of a ticket based on how many tickets have been
sold, and thus on the odds of a split, and the consequent value
of the jackpot:

Business Insider

The horizontal axis shows the number tickets sold, in millions,
and the vertical axis shows the expected value in dollars of a
ticket, based on the likelihood of a split, and the size of a
split jackpot. The blue curve shows the outcome for taking the
annuity prize, and the red line shows the outcome for taking the
cash prize.

Here, we can see that the expected value of a ticket stays
positive (meaning you should consider buying a ticket) as long as
fewer than 730 or so million tickets have been sold. The cash
prize fares far worse — you run into expected losses with only
265 million tickets sold.

So, as long as there are fewer than 730 million tickets sold, a
fairly likely situation right now, the expected value of a ticket
should be positive, and so you should consider buying a Mega
Millions ticket today.

Bear in mind that there are many caveats to this analysis. Taxes
will likely hurt your expected winnings pretty severely — the
Feds will take about 40%, and your home state will claim anywhere
from 0% to around 13%.

A lot of people have been buying tickets, and as discussed above,
this will greatly increase the odds of a tie, and the reduced
payout that goes with it.

The value of the annuity opposed to the cash payout can also be a
complicated question — it is certainly possible that one could
take $341 million and, through prudent investment, end up with a
better 30 year return than the total advertised value of the $636
million annuity.

Given the questionable returns and extreme variance of the
lottery, while the expected value of a Mega Millions ticket might
be positive, the more cautious investor may want to consider
instead adding that dollar to his IRA for a much smaller,
although safer, return. Of course, that is a far less fun option
than going out and writing your six favorite numbers on a ticket.